3.444 \(\int \frac{\sqrt{a d e+(c d^2+a e^2) x+c d e x^2}}{x^4 (d+e x)} \, dx\)

Optimal. Leaf size=286 \[ -\frac{\left (c d^2-a e^2\right ) \left (5 a^2 e^4+2 a c d^2 e^2+c^2 d^4\right ) \tanh ^{-1}\left (\frac{x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{16 a^{5/2} d^{7/2} e^{5/2}}+\frac{\left (3 c d^2-5 a e^2\right ) \left (3 a e^2+c d^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{24 a^2 d^3 e^2 x}-\frac{\left (\frac{c}{a e}-\frac{5 e}{d^2}\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{12 x^2}-\frac{\sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 d x^3} \]

[Out]

-Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(3*d*x^3) - ((c/(a*e) - (5*e)/d^2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x
+ c*d*e*x^2])/(12*x^2) + ((3*c*d^2 - 5*a*e^2)*(c*d^2 + 3*a*e^2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/
(24*a^2*d^3*e^2*x) - ((c*d^2 - a*e^2)*(c^2*d^4 + 2*a*c*d^2*e^2 + 5*a^2*e^4)*ArcTanh[(2*a*d*e + (c*d^2 + a*e^2)
*x)/(2*Sqrt[a]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])])/(16*a^(5/2)*d^(7/2)*e^(5/2))

________________________________________________________________________________________

Rubi [A]  time = 0.404314, antiderivative size = 286, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {849, 834, 806, 724, 206} \[ -\frac{\left (c d^2-a e^2\right ) \left (5 a^2 e^4+2 a c d^2 e^2+c^2 d^4\right ) \tanh ^{-1}\left (\frac{x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{16 a^{5/2} d^{7/2} e^{5/2}}+\frac{\left (3 c d^2-5 a e^2\right ) \left (3 a e^2+c d^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{24 a^2 d^3 e^2 x}-\frac{\left (\frac{c}{a e}-\frac{5 e}{d^2}\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{12 x^2}-\frac{\sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 d x^3} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(x^4*(d + e*x)),x]

[Out]

-Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(3*d*x^3) - ((c/(a*e) - (5*e)/d^2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x
+ c*d*e*x^2])/(12*x^2) + ((3*c*d^2 - 5*a*e^2)*(c*d^2 + 3*a*e^2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/
(24*a^2*d^3*e^2*x) - ((c*d^2 - a*e^2)*(c^2*d^4 + 2*a*c*d^2*e^2 + 5*a^2*e^4)*ArcTanh[(2*a*d*e + (c*d^2 + a*e^2)
*x)/(2*Sqrt[a]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])])/(16*a^(5/2)*d^(7/2)*e^(5/2))

Rule 849

Int[((x_)^(n_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_))/((d_) + (e_.)*(x_)), x_Symbol] :> Int[x^n*(a/d + (c*
x)/e)*(a + b*x + c*x^2)^(p - 1), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b
*d*e + a*e^2, 0] &&  !IntegerQ[p] && ( !IntegerQ[n] ||  !IntegerQ[2*p] || IGtQ[n, 2] || (GtQ[p, 0] && NeQ[n, 2
]))

Rule 834

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[1/((m
+ 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[(c*d*f - f*b*e + a*e*g)*(m + 1)
+ b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] &&
NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ
[2*m, 2*p])

Rule 806

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((e*f - d*g)*(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1))/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2)), x] - Dist[(b
*(e*f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x],
x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && EqQ[Sim
plify[m + 2*p + 3], 0]

Rule 724

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x^4 (d+e x)} \, dx &=\int \frac{a e+c d x}{x^4 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx\\ &=-\frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 d x^3}-\frac{\int \frac{-\frac{1}{2} a e \left (c d^2-5 a e^2\right )+2 a c d e^2 x}{x^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{3 a d e}\\ &=-\frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 d x^3}-\frac{\left (\frac{c}{a e}-\frac{5 e}{d^2}\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 x^2}+\frac{\int \frac{-\frac{1}{4} a e \left (3 c d^2-5 a e^2\right ) \left (c d^2+3 a e^2\right )-\frac{1}{2} a c d e^2 \left (c d^2-5 a e^2\right ) x}{x^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{6 a^2 d^2 e^2}\\ &=-\frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 d x^3}-\frac{\left (\frac{c}{a e}-\frac{5 e}{d^2}\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 x^2}+\frac{\left (3 c d^2-5 a e^2\right ) \left (c d^2+3 a e^2\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 a^2 d^3 e^2 x}+\frac{\left (\left (c d^2-a e^2\right ) \left (c^2 d^4+2 a c d^2 e^2+5 a^2 e^4\right )\right ) \int \frac{1}{x \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{16 a^2 d^3 e^2}\\ &=-\frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 d x^3}-\frac{\left (\frac{c}{a e}-\frac{5 e}{d^2}\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 x^2}+\frac{\left (3 c d^2-5 a e^2\right ) \left (c d^2+3 a e^2\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 a^2 d^3 e^2 x}-\frac{\left (\left (c d^2-a e^2\right ) \left (c^2 d^4+2 a c d^2 e^2+5 a^2 e^4\right )\right ) \operatorname{Subst}\left (\int \frac{1}{4 a d e-x^2} \, dx,x,\frac{2 a d e-\left (-c d^2-a e^2\right ) x}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{8 a^2 d^3 e^2}\\ &=-\frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{3 d x^3}-\frac{\left (\frac{c}{a e}-\frac{5 e}{d^2}\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{12 x^2}+\frac{\left (3 c d^2-5 a e^2\right ) \left (c d^2+3 a e^2\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{24 a^2 d^3 e^2 x}-\frac{\left (c d^2-a e^2\right ) \left (c^2 d^4+2 a c d^2 e^2+5 a^2 e^4\right ) \tanh ^{-1}\left (\frac{2 a d e+\left (c d^2+a e^2\right ) x}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{16 a^{5/2} d^{7/2} e^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.256527, size = 210, normalized size = 0.73 \[ \frac{\sqrt{(d+e x) (a e+c d x)} \left (\frac{\sqrt{a} \sqrt{d} \sqrt{e} \left (a^2 e^2 \left (-8 d^2+10 d e x-15 e^2 x^2\right )-2 a c d^2 e x (d-2 e x)+3 c^2 d^4 x^2\right )}{x^3}-\frac{3 \left (3 a^2 c d^2 e^4-5 a^3 e^6+a c^2 d^4 e^2+c^3 d^6\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a e+c d x}}{\sqrt{a} \sqrt{e} \sqrt{d+e x}}\right )}{\sqrt{d+e x} \sqrt{a e+c d x}}\right )}{24 a^{5/2} d^{7/2} e^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(x^4*(d + e*x)),x]

[Out]

(Sqrt[(a*e + c*d*x)*(d + e*x)]*((Sqrt[a]*Sqrt[d]*Sqrt[e]*(3*c^2*d^4*x^2 - 2*a*c*d^2*e*x*(d - 2*e*x) + a^2*e^2*
(-8*d^2 + 10*d*e*x - 15*e^2*x^2)))/x^3 - (3*(c^3*d^6 + a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 - 5*a^3*e^6)*ArcTanh[(S
qrt[d]*Sqrt[a*e + c*d*x])/(Sqrt[a]*Sqrt[e]*Sqrt[d + e*x])])/(Sqrt[a*e + c*d*x]*Sqrt[d + e*x])))/(24*a^(5/2)*d^
(7/2)*e^(5/2))

________________________________________________________________________________________

Maple [B]  time = 0.069, size = 1165, normalized size = 4.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/x^4/(e*x+d),x)

[Out]

1/4/d/a^2/e^2/x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*c-1/2/d^2/a^2/e/x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^
(3/2)*c-1/16*d^3/a^2/e^2/(a*d*e)^(1/2)*ln((2*a*d*e+(a*e^2+c*d^2)*x+2*(a*d*e)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*
e*x^2)^(1/2))/x)*c^3+11/8/d^3/a*e^2*c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x+1/8*d/a^3/e^2*c^3*(a*d*e+(a*e^
2+c*d^2)*x+c*d*e*x^2)^(1/2)*x-1/2*e^5/d^4*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(d*e*c)^(1/2)+(a*d*e+(a*e^2+c*d^2)*
x+c*d*e*x^2)^(1/2))/(d*e*c)^(1/2)*a+1/2*e^3/d^2*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(d*e*c)^(1/2)+(a*d*e+(a*e^2+c
*d^2)*x+c*d*e*x^2)^(1/2))/(d*e*c)^(1/2)*c+5/16*e^4/d^3*a/(a*d*e)^(1/2)*ln((2*a*d*e+(a*e^2+c*d^2)*x+2*(a*d*e)^(
1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/x)+e^3/d^4*(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2)+3/8*e^3
/d^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+1/2*e^5/d^4*ln((1/2*a*e^2-1/2*c*d^2+(d/e+x)*c*d*e)/(d*e*c)^(1/2)+
(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2))/(d*e*c)^(1/2)*a-1/2*e^3/d^2*ln((1/2*a*e^2-1/2*c*d^2+(d/e+x)*c*d
*e)/(d*e*c)^(1/2)+(c*d*e*(d/e+x)^2+(a*e^2-c*d^2)*(d/e+x))^(1/2))/(d*e*c)^(1/2)*c-1/8/a^3/e^3/x*(a*d*e+(a*e^2+c
*d^2)*x+c*d*e*x^2)^(3/2)*c^2+9/8/d^2/a*e*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*c+1/8*d^2/a^3/e^3*(a*d*e+(a*e
^2+c*d^2)*x+c*d*e*x^2)^(1/2)*c^3-3/16/d*e^2/(a*d*e)^(1/2)*ln((2*a*d*e+(a*e^2+c*d^2)*x+2*(a*d*e)^(1/2)*(a*d*e+(
a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/x)*c-1/16*d/a/(a*d*e)^(1/2)*ln((2*a*d*e+(a*e^2+c*d^2)*x+2*(a*d*e)^(1/2)*(a*d*
e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/x)*c^2+1/2/d/a^2*c^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x-1/3/d^2/a/e
/x^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)-11/8/d^4/a*e/x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+3/4/d^3/a/
x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+3/8/a^2/e*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*c^2

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}{{\left (e x + d\right )} x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/x^4/(e*x+d),x, algorithm="maxima")

[Out]

integrate(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/((e*x + d)*x^4), x)

________________________________________________________________________________________

Fricas [A]  time = 15.7016, size = 1169, normalized size = 4.09 \begin{align*} \left [-\frac{3 \,{\left (c^{3} d^{6} + a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - 5 \, a^{3} e^{6}\right )} \sqrt{a d e} x^{3} \log \left (\frac{8 \, a^{2} d^{2} e^{2} +{\left (c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} x^{2} + 4 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (2 \, a d e +{\left (c d^{2} + a e^{2}\right )} x\right )} \sqrt{a d e} + 8 \,{\left (a c d^{3} e + a^{2} d e^{3}\right )} x}{x^{2}}\right ) + 4 \,{\left (8 \, a^{3} d^{3} e^{3} -{\left (3 \, a c^{2} d^{5} e + 4 \, a^{2} c d^{3} e^{3} - 15 \, a^{3} d e^{5}\right )} x^{2} + 2 \,{\left (a^{2} c d^{4} e^{2} - 5 \, a^{3} d^{2} e^{4}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}{96 \, a^{3} d^{4} e^{3} x^{3}}, \frac{3 \,{\left (c^{3} d^{6} + a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - 5 \, a^{3} e^{6}\right )} \sqrt{-a d e} x^{3} \arctan \left (\frac{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (2 \, a d e +{\left (c d^{2} + a e^{2}\right )} x\right )} \sqrt{-a d e}}{2 \,{\left (a c d^{2} e^{2} x^{2} + a^{2} d^{2} e^{2} +{\left (a c d^{3} e + a^{2} d e^{3}\right )} x\right )}}\right ) - 2 \,{\left (8 \, a^{3} d^{3} e^{3} -{\left (3 \, a c^{2} d^{5} e + 4 \, a^{2} c d^{3} e^{3} - 15 \, a^{3} d e^{5}\right )} x^{2} + 2 \,{\left (a^{2} c d^{4} e^{2} - 5 \, a^{3} d^{2} e^{4}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}{48 \, a^{3} d^{4} e^{3} x^{3}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/x^4/(e*x+d),x, algorithm="fricas")

[Out]

[-1/96*(3*(c^3*d^6 + a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 - 5*a^3*e^6)*sqrt(a*d*e)*x^3*log((8*a^2*d^2*e^2 + (c^2*d^
4 + 6*a*c*d^2*e^2 + a^2*e^4)*x^2 + 4*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*a*d*e + (c*d^2 + a*e^2)*x)
*sqrt(a*d*e) + 8*(a*c*d^3*e + a^2*d*e^3)*x)/x^2) + 4*(8*a^3*d^3*e^3 - (3*a*c^2*d^5*e + 4*a^2*c*d^3*e^3 - 15*a^
3*d*e^5)*x^2 + 2*(a^2*c*d^4*e^2 - 5*a^3*d^2*e^4)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(a^3*d^4*e^3*
x^3), 1/48*(3*(c^3*d^6 + a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 - 5*a^3*e^6)*sqrt(-a*d*e)*x^3*arctan(1/2*sqrt(c*d*e*x
^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*a*d*e + (c*d^2 + a*e^2)*x)*sqrt(-a*d*e)/(a*c*d^2*e^2*x^2 + a^2*d^2*e^2 + (a
*c*d^3*e + a^2*d*e^3)*x)) - 2*(8*a^3*d^3*e^3 - (3*a*c^2*d^5*e + 4*a^2*c*d^3*e^3 - 15*a^3*d*e^5)*x^2 + 2*(a^2*c
*d^4*e^2 - 5*a^3*d^2*e^4)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x))/(a^3*d^4*e^3*x^3)]

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2)/x**4/(e*x+d),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)/x^4/(e*x+d),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError